1. Signals and Systems Lecture 9
Fourier Series and LTI Systems
Frequency Response and Filtering

2. Chapter 3 Fourier Series
§ Fourier Series and LTI Systems
Eigenfunctions of LTI System
System Function
System Function

3. Chapter 3 Fourier Series
§ Fourier Series and LTI Systems
Frequency Response of an LTI System
Frequency Response
of CT LTI Systems
Frequency Response
of DT LTI Systems

4. Chapter 3 Fourier Series
§ Fourier Series and LTI Systems
Continuous-time LTI System
Linear Combinations
of Eigenfunctions
Periodic Signal
Frequency Response
of LTI System
including both amplitude & phase
must be well defined and finite.

5. Example:3.16 Chapter 3 Fourier Series Input signal: Impulse response:
Find out bk of output signal
Frequency response:

6. Chapter 3 Fourier Series
Discrete-time LTI System
Linear Combinations
of Eigenfunctions
Periodic Signal
Frequency Response
of LTI System
including both amplitude & phase
must be well defined and finite.

7. Example:3.17 Chapter 3 Fourier Series Input signal: Impulse response:
Find out the output signal
Frequency response:
If N=4, we get
then
and

8. Chapter 3 Fourier Series
Example
Consider an LTI system with input
the unit impulse response , determine the
Fourier Series Representation of output
k is even
k is odd

9. Frequency-Shaping Filter Frequency-Selective Filter
Chapter Fourier Series
§ Filtering（滤波）
Filter
Frequency-Shaping Filter
频率成形滤波器
Frequency-Selective Filter
频率选择性滤波器

10. Chapter 3 Fourier Series
§3.9.1 Frequency-Shaping Filter
Change the shape of the spectrum
1. Equalizer（均衡器）

11. Chapter 3 Fourier Series
2. Differentiator （微分器） D

12. Chapter 3 Fourier Series
Figure 3.24

13. Chapter 3 Fourier Series
3. Discrete-time system

14. Chapter 3 Fourier Series
§3.9.2 Frequency-Selective Filter
Filter out signals outside of the frequency range of interest
Lowpass Filters:
Only show amplitude here.
Note for DT:

15. Chapter 3 Fourier Series
§3.9.2 Frequency-Selective Filter
Highpass Filters
Remember: — = highest frequency in DT

16. Chapter 3 Fourier Series
§3.9.2 Frequency-Selective Filter
Bandpass Filters

17. Chapter 3 Fourier Series
§3.9.2 Frequency-Selective Filter
1. Ideal Lowpass Filter (LPF)
Continuous-time system
Stopband
Passband
Discrete-time system
Stopband
Passband

18. Chapter 3 Fourier Series
2. Ideal Highpass Filter (HPF)
Continuous-time system
Passband
Stopband
Discrete-time system

19. Chapter 3 Fourier Series
3. Ideal Bandpass Filter (BPF)
Continuous-time system
Discrete-time system

20. Chapter 3 Fourier Series Example Consider an LTI system with input
the frequency response of this system is
as shown in Figure 1 , determine the output of system
Figure 1 (a)
Figure 1 (b)

21. Chapter 3 Fourier Series
§3.10 Examples of continuous-time filters described
by differential equations
§ A simple RC Lowpass Filter

22. Chapter 3 Fourier Series
§ A simple RC Highpass Filter

23. Chapter 3 Fourier Series
§3.11 Examples of discrete-time filters described
by difference equations
§ First-Order Recursive DT Filter

24. Chapter 3 Fourier Series
§ Nonrecursive DT Filters Example :
DT Averager/Smoother

25. Chapter 3 Fourier Series
§ Nonrecursive DT Filters Example :
DT Averager/Smoother
Rolls off at nω lower as M+N+1 increases

26. Example : Simple DT “Edge” Detector
Chapter Fourier Series
Example : Simple DT “Edge” Detector
DT 2-points “differentiator”
Amplifies high-frequency components

27. Summary Fourier Series and LTI Systems
Frequency Response and Filtering

28. Readlist
Signals and Systems:
Chapter1~Chapter3

29. Chapter 3 Fourier Series
Homework:

30. Chapter 3 Fourier Series
Homework:

31. Chapter 3 Fourier Series
Eigenfunction
Eigenvalue

32. Synthesis equation
综合公式
Analysis equation
分析公式
Chapter Fourier Series
1. Periodic square wave
2. Periodic Impulse Trains

33. Chapter 3 Fourier Series

34. Chapter 3 Fourier Series
real odd
Purely imaginary odd
real even